FROM THREEVALUED NESTED SETS TO INTERVALVALUED (OR INTUITIONISTIC) FUZZY SETS


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1 INTERNATIONAL JOURNAL OF INFORMATION AND SYSTEMS SCIENCES Volume 7, Number 1, Pages c 2011 Institute for Scientific Computing and Information FROM THREEVALUED NESTED SETS TO INTERVALVALUED (OR INTUITIONISTIC) FUZZY SETS CHENG ZHANG, QING SU, ZHIWU ZHAO, AND PANZI XIAO Abstract. In this paper,we first set up the equivalence class of threevalued nested sets based on threevalued fuzzy sets, and the operations of equivalence classes are derived from the ones of threevalued fuzzy sets. We then prove that the intervalvalued (or intuitionistic) fuzzy set can be regarded as the equivalence class of threevalued nested set. Finally, we apply the above results to the research of the intervalvalued convex fuzzy set. In addition, the above outcomes also show the correctness of the theory of intervalvalued (/or intuitionistic) fuzzy set established by the paper [1]. Key Words. threevalued fuzzy set, threevalued nested set, intuitionistic fuzzy set, intervalvalued fuzzy set. 1. Introduction In the 70s last century, the notion of a intervalvalued fuzzy set has been introduced by different authors(see[2], [3] and[4])in order to formalize the vagueness. In recent years, the researches of intervalvalued fuzzy set are growing, that is because in the course of applying to practice, the membership function of a fuzzy set is difficult to be determined, but the membership degree of an intervalvalue is relatively easy to be determined. Besides, the results of fuzzy inference with intervalvalued fuzzy sets can be better to reflect the ambiguity of everyday reasoning. In 1985, the interval representation of language value was discussed by Schwarz in [5]. In 1986, intervalvalued fuzzy sets which based on the norm were studied by Turksen in [6]. In 1987, a method about intervalvalued fuzzy inference was given by Gorzalczany in [7]. In the paper [8][11], the basic research of intervalvalued fuzzy sets were studied. In 1986, the notion of intuitionistic fuzzy sets were introduced by Atanassov in [12], from then on, the intuitionistic fuzzy groups, intuitionistic fuzzy subspaces and intuitionistic fuzzy topology appeared in [13][16]. Although the results on the intervalvalued (/or intuitionistic) fuzzy set were richness, the nested sets theory on the intervalvalued (/or intuitionistic) fuzzy set is not established similarly to the fuzzy sets, the reason lied in the intervalvalued fuzzy sets cut sets which defined in the article [8][10]was a classical set. The similar case appears in the theory of the intuitionistic fuzzy sets, which leads to nested sets theory for the intervalvalued (/or intuitionistic) fuzzy sets that have not been established so far. We know that a fuzzy set can be seen as a equivalence class of a nested set [17]. In the fuzzy sets theory, we can define a fuzzy set according to a equivalence class of classical nested set, and the operations of the fuzzy sets according to the Received by the editors June 10, 2010 and, in revised form, July 22, Mathematics Subject Classification. 93A30. This research was supported in part by National Natural Science Foundation of China (Major Research Plan, No ). 11
2 12 C. ZHANG, Q. SU, Z. ZHAO AND P. XIAO operations and, or, not of classical sets. What s more, we can also obtain the definition of the membership function of the fuzzy sets via characteristic function. In this manner, we accomplish the extension from boolean algebra to soft algebra. Meanwhile, a problem occurs: whether a intervalvalued fuzzy set can be seen as a equivalence class of some nested sets. Therefore, we begin our work for the interest of this problem. The paper is organized as follows. In section 2, we present threevalued nested sets and their operations. In section 3, we establish equivalence classes of threevalued nested sets and present their operations. In section 4, we construct a isomorphism between the intervalvalued (/or intuitionistic) fuzzy set system and the threevalued nested set system. In section 5, we put the above theory into intervalvalued (/or intuitionistic) convex fuzzy sets research. 2. Threevalued nested sets and its operation Let X be a set. The mapping A : X {0, 1 2,1} is called a threevalued fuzzy set on X. A class of all threevalued fuzzy sets over X is denoted by 3 X. For any A,B,A t 3 X,x X, the operations are defined as follows: (A B)(x) = A(x) B(x); (A B(x) = A(x) B(x); (ca)(x) = 1 A(x); ( A t )(x) = A t (x); ( A t )(x) = A t (x); X(x) 1, x X; (x) 0, x X. Then (3 X,,,C,X, )is an F lattice. Definition 2.1. [1] A mapping H : [0,1] 3 X,λ H(λ) is called a threevalued nested set of X if H satisfies that H(λ 1 ) H(λ 2 ), whenever λ 1,λ 2 [0,1] and λ 1 < λ 2. We use IU denote the class of all the threevalued nested sets over X. There exist maximum element X and minimum element,x(λ) X, (λ), which satisfy X H = X,X H = H, H = H, H =. Definition 2.2. [1] For H i IU(X)(i = 1,2),H t IU(X)(t T), the operations are defined as follows: H 1 H 2 : (H 1 H 2 )(λ) = H 1 (λ) H 2 (λ); H 1 H 2 : (H 1 H 2 )(λ) = H 1 (λ) H 2 (λ); CH : (CH)(λ) = CH(1 λ);
3 FROM THREEVALUED NESTED SETS TO IVFS 13 H t : ( H t )(λ) = H t (λ); H t : ( H t )(λ) = H t (λ). It is not difficult to verify that H 1 H 2,H 1 H 2,CH, threevalued nested sets. H t and Proposition 2.1. (IU,,,C,X, ) has the properties as follows: (1) Idempotent law: H H = H,H H = H; (2) Commutative law: H 1 H 2 = H 2 H 1,H 1 H 2 = H 2 H 1 ; (3) Associative law: H 1 (H 2 H 3 ) = (H 1 H 2 ) H 3,H 1 (H 2 H 3 ) = (H 1 H 2 ) H 3 ; (4) Absorption law: H 1 (H 1 H 2 ) = H 1 (H 1 H 2 ) = H 1 ; (5) Distributive law: H 1 (H 2 H 3 ) = (H 1 H 2 ) (H 1 H 3 ), H 1 (H 2 H 3 ) = (H 1 H 2 ) (H 1 H 3 ); (6) Identity law: H X = H,H = H,H X = X,H = ; (7) Involution law: C(CH) = H; (8) De Morgan law: C( H t ) = CH t,c( H t ) = CH t ; (9) Infinitedistribute law: H ( H t ) = (H H t ),H ( H t ) = H t are also (H H t ). Proof. We only show that (8) and (9) are correct, and the others are similar. (C( H t ))(λ) = C(( H t )(1 λ)) = C( H t (1 λ)) Thus, C( H t ) = CH t. = C(H t (1 λ)) = (CH t )(λ). (H ( H t ))(λ) = H(λ) ( H t )(λ) = H(λ) ( H t (λ)) Therefore, H ( H t ) = (H H t ). = (H(λ) H t (λ)) = (H H t )(λ). The others are parallelism, and proofs are omitted. By Proposition 2.1, we have known that threevalued nested sets have similar properties to the fuzzy sets. 3. Equivalence class of threevalued nested sets Definition 3.1. A relation in IU(X) is defined as follows: for any λ (0,1]. H 1 H 2 H 1 (α) = H 2 (α)
4 14 C. ZHANG, Q. SU, Z. ZHAO AND P. XIAO Then the above is an equivalence relation, that is, it satisfies (a) reflexivity: H H; (b) symmetry: H 1 H 2 H 2 H 1 ; (c) transitivity: H 1 H 2,H 2 H 3 H 1 H 3. In IU(X), we define for all λ [0,1]. Let and H 1 H 2 H 1 (λ) H 2 (λ) [H] = {H H U(X),H H} IVF (X) IU(X)/ = {[H] H IU(X)} Lemma 3.1. Let H IU(X), we define F H : [0,1] 3 X,λ F H (λ) H(α),(λ (0,1]),F H (0) = X; F H : [0,1] 3 X,λ F H (λ) H(α),(λ [0,1)),F H (1) =. Then (1) F H,F H IU(X); (2) λ 1 < λ 2 F H (λ 1 ) F H (λ 2 ); (3) F H H F H ; (4) F H (λ) = F H (α),(λ (0,1]),F H (λ) = (5) F H (λ) = F H (α),(λ (0,1]),F H (λ) = F H (α),(λ [0,1)); F H (α),(λ [0,1)). Proof. (1) If λ 1 < λ 2, then F H (λ 1 ) = H(α) H(α) = F H (λ 2 ). 1 2 (2) If λ 1 < λ 2, then there exists α 0 such that λ 1 < α 0 < λ 2, thus F H (λ 1 ) = H(α) H(α 0 ) H(α) = F H (λ 2 ). 1 2 (3) By H(α) H(λ)( α > λ), we have F H (λ) = By H(α) H(λ)( α < λ), we have F H (λ) = It follows that therefore (4) Since then F H (λ) = F H (α) = F H (λ) H(λ) F H (λ), β<α F H (α),(λ (0,1]). In the following, we will show that If λ 1, then F H (α) F H (λ) F H H F H. H(β) = β<α, F H (λ) = F H (α),(λ [0,1)). hand, F H (α) H(α) for all α [0,1], thus follows that F H (λ) = F H (α). (5) we first prove that F H (λ) = H(α) H(λ). H(α) H(λ). H(β) = β<λ H(β) = F H (λ), F H (λ) F H (λ) for all α > λ. On the other F H (α) H(α) = F H (λ), it F H (α),(λ (0,1]).
5 FROM THREEVALUED NESTED SETS TO IVFS 15 If λ 0, then by F H (α) F H (λ) for any α < λ, we have On the other hand, F H (α) H(α) for any α [0,1], it follows that H(α) = F H (λ). Therefore F H (λ) = In the next place, we will prove that F H (λ) = In fact, F H (α) = That is, H(β) = β>α β>α,α>β Lemma 3.2. Let H 1,H 2 IU(X), then H 1 H 2 Proof. It only needs to prove that F H (α)(λ (0,1]). F H (α) F H (λ). F H (α)(λ [0,1)). F H (α) H(β) = H(β) = F H (λ)(λ [0,1)). β>λ F H (λ) = F H (α). H 1 (α) = H 2 (α)(λ [0,1)) F H1 (λ) = F H2 (λ) λ [0,1),F H1 (λ) = F H2 (λ) for all λ (0,1]. As a matter of fact, F H1 (λ) = F H2 (λ) F H1 (λ) = (0,1]; F H1 (λ) = F H2 (λ) F H1 (λ) = F H1 (α) = F H1 (α) = F H2 (α) = F H2 (λ) for all λ F H2 (α) = F H2 (λ) for all λ [0,1). Theorem 3.1. Let H,H,H t,h t IU(X)(t T) and H t H t (t T),H H, then H t H t H t,ch CH. H t, Proof. (1) According to H t H t(t T) H t (α) = H t (α)( λ (0,1]), it is obvious that in order to have that H t we only need to prove that H t ( = )(α) ( H t)(α). In fact, ( H t )(α) = H t (α) = H t (α) = H t (α) = H t (α) = ( H t )(α)(λ (0,1]). (2) By Lemma 3.2, we only need to prove that ( H t )(α) = ( H t)(α) Asamatteroffact, ( H t )(α) = H t (α)= H t (α) = H t (α) = H t (α) = ( H t )(α). (3) By lemma 3.2, H t H t(t T) H t (α) = H t(α)(λ [0,1)).
6 16 C. ZHANG, Q. SU, Z. ZHAO AND P. XIAO (CH)(α) = CH(1 α) = C( H(1 α)) = C( 1 α>1 λ = C(H (1 α)) = (CH )(α),(λ [0,1)). By Definition 3.1, we known that CH CH. 1 α>1 λ Definition 3.2. The operations,,c in IVF (X) are defined as follows: [H t ] = [ H t ], [H t ] = [ H t ],C[H] = [CH]. By Theorem 3.1, we have known that Definition 3.2 is welldefined. For A 3 X, let H A (λ) A,IP (X) = {[H A ] A 3 X }, H (1 α)) then IP (X) IVF (X),(IP (X),,,C) and (3 X,,,C) is isomorphism. Let IVF(X) = (IVF (X) IP (X)) 3 X, then we obtain a new algebra system (IVF(X),,,C), and it is isomorphic with system (IVF (X),,,C) by the embedding theorem of algebra [18]. In the following, wewillprovethat the algebrasystem (IVF(X),,,C) canbe isomorphic with the algebra system which formed by intervalvalued fuzzy sets (/or intuitionistic) fuzzy sets. So we can regard the algebra system (IVF(X),,,C) as the model of intervalvalued (/or intuitionist) fuzzy sets model. 4. Membership functions of intervalvalued(/or intuitionistic)fuzzy sets Let L = {(α,β) α,β [0,1],α β}, L X = {A A : X L is a mapping}. A L X is called a intervalvalued fuzzy set and denoted by A = [A,A + ], which A : X [0,1],A + : X [0,1], moreover A (X) A + (X), x X. Definition 4.1. For any A,B,A (γ) L X, let A = [A,A + ],B = [B,B + ],A (γ) = [A (γ),a+ (γ)], we define A B A (x) B (X),A + (X) B + (X), x X; A = B A (X) = B (X),A + (X) = B + (X), x X; CA = [1 A +,1 A ]; (γ) = γ ΓA [ γ ΓA (γ), A + (γ) ]; γ Γ (γ) = γ ΓA [ γ ΓA (γ), A + (γ) ]. γ Γ Theorem 4.1. If mapping f : IVF(X) L X,A = [H] [A,A + ]. where A (x) = {λ H(λ)(x) = 1},A + (x) = {λ H(λ)(x) 1 2 }. Then (1) f is a bijective. (2) A λ = {x F H(λ)(x) = 1},A + λ = {x F H(λ)(x) 1 2 }, A λ = {x F H(λ)(x) = 1},A + λ = {x F H(λ)(x) 1 2 }. (3) f keeps operations, that is f( [H t ]) = f([h t ]),f( t ]) = [H f([h t ]),f(c[h]) = Cf([H])
7 FROM THREEVALUED NESTED SETS TO IVFS 17 Proof. (1) Firstly, we need to prove that f is welldefined. Suppose that A = [H] IVF(X). By Lemma 3.1, for H [H] we have It follows that F H (λ) = F H (λ) H (λ) F H (λ) = F H (λ)( λ [0,1]). {λ F H (λ)(x) = 1} {λ H(λ)(x) = 1} {λ F H (λ)(x) = 1}, {λ F H (λ)(x) 1 2 } {λ H(λ)(x) 1 2 } {λ F H(λ) 1 2 }. For any λ {α F H (α)(x) = 1}, we have F H (λ )(x) = 1. Since F H (λ) F H (λ ) for λ < λ, thus F H (λ)(x) F H (λ )(x), and then F H (λ)(x) = 1, therefore, {λ F H (λ)(x) = 1} {λ λ [0,1],λ < λ } = λ, consequently {λ F H (λ)(x) = 1} {α F H (α)(x) = 1}, therefore {λ F H (λ)(x) = 1} = {λ F H (λ)(x) = 1}. Analogously, λ {α F H (α)(x) 1 2 }, we have F H(λ )(x) 1 2 for any λ < λ, and then F H (λ) F H (λ ), so F H (λ)(x) F H (λ )(x) 1 2, thus {λ F H (λ)(x) 1 2 } {λ λ [0,1],λ < λ } = λ, which implies that {λ F H (λ)(x) 1 2 } {α F H (α)(x) 1 2 }. It follows that {λ F H (λ)(x) 1 2 } = {λ F H(λ)(x) 1 2 }. Therefore f is welldefined, and furthermore, A (x) = {λ F H (λ)(x) = 1} = {λ H(λ)(x) = 1} = {λ F H (λ)(x) = 1}, A + (x) = {λ F H (λ)(x) 1 2 } = {λ H(λ)(x) 1 2 } = {λ F H(λ)(x) 1 2 }. (2) If F H (λ)(x) = 1, then A (x) = {λ F H (λ)(x) = 1} λ; If F H (λ)(x) 1, then F H (λ)(x) = ( H(α))(x) = H(α)(x) 1, thus there exists α 0 < λ, such that H(α 0 )(x) 1. For all λ α 0, we have H(λ ) H(α 0 ), then H(λ )(x) 1, so {λ H(λ)(x) = 1} {λ λ α 0 }, this leads to a contradiction which implies that A (x) = {λ H(λ)(x) = 1} {λ λ α 0 } = α 0 < λ, A λ = {x F H(λ)(x) = 1}. Similarly, we have that A + λ = {x F H(λ)(x) 1 2 },A λ = {x F H (λ)(x) = 1} and A + λ = {x F H (λ)(x) 1 2 }. (3) We will prove that f is to be a bijective. On the one hand, suppose A L X, put 1, x A λ 1 H(λ)(x) = 2, x A+ λ A λ 0, x / A + λ then H(λ) is a threevalued fuzzy sets. According to A (x) = {λ H(λ)(x) = 1} = {λ x A λ },A+ (x) = {λ H(λ)(x) 1 2 } = {λ x A+ λ }, we know that f([h]) = A = [A,A + ], thus f is a surjection. On the other hand, if f([h]) = f([h ]) = A = [A,A + ], then A (x) = {λ H(λ)(x) = 1} = {λ H (λ)(x) = 1},
8 18 C. ZHANG, Q. SU, Z. ZHAO AND P. XIAO Thus for any x X, we have A + (x) = {λ H(λ)(x) 1 2 } = {λ H (λ)(x) 1 2 }. F H (λ)(x) = 1 x A λ F H (λ)(x) = 1,F H(λ)(x) = 0 x / A + λ F H (λ)(x) = 0. Therefore H(α) = F H (λ) = F H (λ) = H (α)(λ [0,1]), i.e., [H] = [H ], which implies that f is a injection. Consequently, f is a bijection. (4) we will prove them respectively. (a) Let f([h t ]) = [A t,a+ t ], f([h]) = [ A t, A + t ], then we have that F H (λ)(x) = 1 x ( A t ) λ t T,F Ht (λ)(x) = 1 ( F Ht )(λ)(x) = 1 and F H (λ)(x) = 0 x / ( A + t ) λ ( A + t )(x) < λ t 0 T,(A + t 0 )(x) < λ F Ht0 (λ)(x) = 0 F Ht (λ)(x) = 0 ( F Ht (λ))(x) = 0. Then F H (λ)(x) = ( F Ht (λ))(x), i.e., F H (λ) = ( F Ht )(λ), thus [H] = [F H ] = [ F Ht ] = [F Ht ] = [H t ], itfollowsthatf( [H t ]) = f([h]) = [ A t, A + t ] = [A t,a+ t ] = f([h t ]). (b)leth,h t IU(X)(t T)andf([H t ]) = [A t,a+ t ],f([h]) = [ A t, A + t ]. Then we have that F H (λ)(x) = 1 x ( A t ) λ t 0 T,F Ht0 (λ)(x) = 1 ( F Ht )(λ)(x) = 1 and F H (λ)(x) = 0 x / ( A + t ) λ A + t (x) λ t T,(A + t )(x) < λ t T,F Ht (λ)(x) = 0 F Ht (λ)(x) = 0 ( F Ht (λ))(x) = 0. Thus F H (λ)(x) = ( F Ht (λ))(x), i.e, F H (λ) = ( F Ht )(λ). Hence [H] = [F H ] = [ F Ht ] = [F Ht ] = [H t ]. It follows that f( [H t ]) = f([h]) = [ A t, A + t ] = [A t,a + t ] = f([h t ]). (c) Let f([h]) = [A,A + ], [H] IU(X) and f([h]) = [CA +,CA ]. Then we get F H (λ)(x) = 1 x (CA + ) λ (CA + )(x) λ 1 A + (x) λ A + (x) 1 λ x / A + 1 λ F H (1 λ)(x) = 0 CF H (1 λ)(x) = 1 (CF H )(λ)(x) = 1 and F H (λ)(x) = 0 x / (CA ) λ (CA )(x) < λ 1 A (x) < λ A (x) > 1 λ x A 1 λ F H (1 λ)(x) = 1 CF H (1 λ)(x) = 0 (CF H )(λ)(x) = 0. Thus F H (λ) = (CF H )(λ), therefore [H] = [F H ] = [CF H ] = C[F H ] = C[H]. It follows that f(c[h]) = f[h] = C(f([H])). By Theorem 4.1, it is obvious that the system (IVF(X),,,C) is isomorphic with the system (L X,,,C). Hence we need not to distinguish A = [H] from f([h]) = [A,A + ], and call [A (x),a + (x)] as the membership interval of x for A. If we consider the intervalvalued fuzzy set A = [H] and the membership interval of A as the same, we can obtain the definition of Zadeh intervalvalued fuzzy sets. Therefore, from the viewpoint of the mathematics, an intervalvalued fuzzy set can be seen as an equivalent class, which illustrates the expansion from threevalued case to intervalvalued fuzzy case. Let L = {(a,b) a,b [0,1],a+b 1)}, we define
9 FROM THREEVALUED NESTED SETS TO IVFS 19 (a) (a 1,b 1 ) (a 2,b 2 ) a 1 a 2,b 1 b 2 ; (b) (a 1,b 1 ) (a 2,b 2 ) = (a 1 a 2,b 1 b 2 ),(a 1,b 1 ) (a 2,b 2 ) = (a 1 a 2,b 1 b 2 ); (c) (a,b) c = (b,a); (d) (a t,b t ) = ( a t, b t ); (a t,b t ) = ( a t, b t ); (e) 1 = (1,0),0 = (0,1). Suppose L X = {A A : X L is a mapping }, and A L X. A is said to be a intuitionistic fuzzy set and denoted by A = (µ A,ν A ), where µ A : X [0,1],ν A : X [0, 1] represent the degree of membership and nonmembership, respectively, satisfying µ A (x)+ν A (x) 1, x X. Let IF(X) denote the class of all the intuitionistic fuzzy sets over X, the operations in IF(X) are derived by the operations in L as follows: Let A = (µ A,ν A ),B = (µ B,ν B ),A t = (µ At,ν At ),(t T), then (1) A B µ A (x) µ B (x),ν A (x) ν B (x), x X; (2) CA = (ν A,µ A ); (3) A B = (µ A B,ν A B ),where µ A B = µ A (x) µ B (x),ν A B = µ A (x) µ B (x); (4) A B = (µ A B,ν A B ),where µ A B = µ A (x) µ B (x),ν A B = µ A (x) µ B (x); (5) Let A = A t, where µ A (x) = µ At (x),ν A (x) = ν At (x); (6) Let B = A t, where µ B (x) = µ At (x),ν B (x) = ν At (x); (7) Let X(x) (1,0), x X; (x) = (0,1), x X. (IF(X),,,C, X, ) is a fuzzy lattice. We define a mapping between IF(X) and IVF(X) by ϕ : A = (µ A,ν A ) [µ A,Cν A ], then the mapping ϕ is a isomorphism of algebraic systems (IF(X),,,C) and (IVF(X),,,C). By the Theorem 4.1, we have Theorem 4.2. If mapping h : IVF(X) L X,A = [H] (µ A,ν A ), where µ A (x) = {λ H(λ)(x) = 1},ν A (x) = {1 λ H(λ)(x) 1 2 }, then (1) h is a bijective; (2) (µ A ) λ = {x F H (λ)(x) = 1},(ν A ) λ = {x F H (1 λ)(x) = 0}, (µ A ) λ = {x F H (λ)(x) = 1},(ν A ) λ = {x F H (1 λ)(x) = 0}. (3) h keeps the operations, i.e., h( [H t ]) = h([h t ]),h( [H t ]) = h([h t ]),h(c[h]) = Ch([H]). 5. Intervalvalued(/or intuitionistic) convex fuzzy sets Suppose X is a vector space, let A = [A,A + ] be a intervalvalued fuzzy set of X. By Theorem 4.1, we have that the algebra system (IVF,,,C) is isomorphic with the algebra system (L X,,,C) (/or (L X,,,C)), therefore, there exists [H] IVF(X) such that A (x) = {λ H(λ)(x) = 1}, A + (x) = {λ H(λ)(x) 1 2 }. A = [A,A + ] is said to be a intervalvalued convex fuzzy set over X if there exists a threevalued nested set H in the equivalence class [H] such that H(λ) is a threevalued convex fuzzy set. Theorem 5.1. The following assertions are equivalent. (1) A = [A,A + ] is an intervalvalued convex fuzzy set in X; (2) A λ anda+ λ are convex sets for arbitrary λ (0,1]; (3) For arbitrary a [0,1] and x,y X, then A (ax+(1 a)y) min{a (x),a (y)},a + (ax+(1 a)y) min{a + (x),a + (y)}; (4) A λ,a+ λ are convex sets for arbitrary λ (0,1];
10 20 C. ZHANG, Q. SU, Z. ZHAO AND P. XIAO (5) For arbitrary x 1,,x p E and all a 1,,a p [0,1] which make p a i = 1, we have A ( p a i x i ) min(a (x 1 ), A (x p )),A + ( p a i x i ) min(a + (x 1 ), A + (x p )). i=1 The similar conclusions can be acquired on intuitionistic fuzzy sets by Theorem 4.2, and we omit them in this paper. 6. Conclusions Through the research of threevalued fuzzy sets and threevalued nested sets in paper[1], the definition the intervalvalued(/or intuitionistic) fuzzy set is formulated. Based on the equivalence class of threevalued fuzzy nested sets and the operations of intervalvalued(/or intuitionistic) fuzzy sets also are formed, which illustrated the expansion from fuzzy sets to intervalvalued(/or intuitionistic) fuzzy sets and supplied the theoretical basis for paper[1] from another aspect. Furthermore, as the application of the above theory, we discussed the intervalvalued(/or intuitionistic) convex fuzzy set, and the derived results show that when the intervalvalued(/or intuitionistic) fuzzy set degenerate the fuzzy set, the above theory is consistent with fuzzy set theory. However, we can not obtain the above ones using the previous concept of the intervalvalued fuzzy set s cut set. It shows that the definition of the cut set defined in the paper [1] is reasonable. By analogy with the fuzzy set, we will try to explore the algebra, analysis, optimization and decision theory of the intervalvalued(/or intuitionistic) fuzzy set. Acknowledgments The author thanks the anonymous authors whose work largely constitutes this sample file. This research was supported by in part by National Natural Science Foundation of China (Major Research Plan, No ). References [1] Yuan Xuehai,Li Hong xing,sun Kaibiao. The cut set of Intuitionistic fuzzy sets and intervalvalued fuzzy sets and decomposition theorem and the Representation Theorem, Science in China (E Series),2009,39(9): [2] Zadeh L A., Outline of a new approach to the analysis of complex systems and decision processes, intervalvalued fuzzy sets, IEEE Trans. Systems Man Cybernet.1973, 3(1): [3] GrattanGuiness, I., Fuzzy membership mapped onto interval and many valued quantities, Z. Math. Logik Grundlag. Math., 22(1975), [4] Jahn, K. U., Intervalwertige mengen. Math. Nachr., 68(1975), [5] Schwarz D. G., The case for an interval based representation of ligiistic truth, Fuzzy Sets and Systems, 20(1985), [6] Turksen I.B., Interval valued fuzzy sets based on normal forms, Fuzzy Sets and Systems, 20(1986), [7] Gorzalczany M.B., A method of inference in approximate reasoning based on intervalvalued fuzzy sets, Fuzzy Sets and Systems, 21(1987), [8] Meng Guangwu,. The Basic Theory of Intervalvalued Fuzzy sets. Applied Mathematics(in Chinese), 6(1993), No.2: [9] Zeng Wenyi,Li Hongxing,Shi Yu. decomposition theorem of Intervalvalued fuzzy sets, Journal of Beijing Normal University(in Chinese), 2003,39(2): [10] Zeng Wenyi,Li Hongxing,Shi Yu. Representation theorem of Intervalvalued fuzzy sets. Journal of Beijing Normal University(in Chinese), 2003,39(4): [11] Glad Deschrijver, Generalized arithmetic operators and their relationship to tnorms in intervalvalued fuzzy set theory, Fuzzy Sets and Systems, 160 (2009) [12] Atanassov K. Intuitionistic fuzzy Sets.Fuzzy Sets and Systems, 1986, 20: [13] Biswas R. Intuitionistic fuzzy subgroups, Math.Fortum, 1989,10: i=1 i=1
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